Explanation:
We know that the vertex of the parabola is at the origin, so the equation must have the form:
y = a x^2
where a is a constant.
We also know that the directrix is y = -1.5, which means that the focus is located at (0, 1.5).
Using the formula for the distance between a point (x, y) and a line ax + by + c = 0, which is given by:
d = |ax + by + c| / sqrt(a^2 + b^2)
we can find the value of a. The distance from the vertex (0, 0) to the directrix y = -1.5 is equal to the distance from the vertex to the focus (0, 1.5), so we have:
d = |y - (-1.5)| / sqrt(1^2 + 0^2) = sqrt(x^2 + (y - 1.5)^2) / sqrt(0^2 + 1^2)
Simplifying this equation, we get:
sqrt(x^2 + (y - 1.5)^2) = |y + 1.5|
Squaring both sides, we get:
x^2 + (y - 1.5)^2 = (y + 1.5)^2
Expanding and simplifying, we get:
x^2 = 3y
So the equation of the parabola with vertex at the origin and directrix y = -1.5 is:
y = (1/3)x^2
Therefore, the correct answer is: y = (1/3)x^2.